Counting - the end.

In my fruit dish there are four apples.  What does that mean?  How do I know?  When we're little we count on our fingers - it's no coincidence that we count in tens, because that's as far as you can go before you run out of fingers and have to start again.  So we learn that if we stick our thumb up, that's called "one" and if we have a set of things which we can put in exact one-to-one correspondence (i.e. none left over) with our thumb, then that set is also called one.  Add our first finger and we now have a set called "two" and any set which can be put in exact one-to-one correspondence with that is also called two.  And so on...  So I know I've got four apples because I can put them in exact one-to-one correspondence with my thumb, first finger, middle finger and ring finger and I know that set is called four.  Of course as we grow up we stop depending on our fingers - we can visualise the numbers - at least the relatively small ones, but the principle remains.  So this series of posts has sort of come full circle - we started out by seeing that "set theory" as it's called can enable you to count without really counting, and we end up by realising that counting is in fact a structured form of set theory.  And that, I think, is enough!